## exercise 1.108 (Greece 2008) [BOMC]

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nafistiham
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### exercise 1.108 (Greece 2008) [BOMC]

can we use anything else than "the helpful inequality"

for $x_{1},x_{2},x_{3},\cdot \cdot \cdot x_{n}$ positive integers, prove that

$(\frac{x_{1}^{2}+x_{2}^{2}+\cdot \cdot \cdot +x_{n}^{2}}{x_{1}+x_{2}+\cdot \cdot \cdot +x_{n}})^{\frac{kn}{t}}\geq x_{1}\cdot x_{2}\cdot \cdot \cdot \cdot \cdot x_{n}$,

where $k=$ max {${{x_{1}, x_{2},\cdot \cdot \cdot ,x_{n}}}$} and $t=$ min {${{x_{1}, x_{2}, \cdot \cdot \cdot, x_{n}}}$}.
under which conditions the equality holds?
$\sum_{k=0}^{n-1}e^{\frac{2 \pi i k}{n}}=0$
Using $L^AT_EX$ and following the rules of the forum are very easy but really important, too.Please co-operate.

*Mahi*
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### Re: exercise 1.108 (Greece 2008) [BOMC]

In fact you can use Cauchy or QM-AM inequalities to simplify this problem.
Then solve the last part as you please.

Use $L^AT_EX$, It makes our work a lot easier!

Hasib
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### Re: exercise 1.108 (Greece 2008) [BOMC]

But, as far as i proved,

$(\frac{\sum_{i=1}^{n} x_i^2}{\sum_{i=1}^{n} x_i})^n \geq x_1\cdot \cdot x_n$

but, how raise the power $\frac{k}{t}$?
A man is not finished when he's defeated, he's finished when he quits.